Abstract
In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We consider many cases, where restricted to a permutation class, the staircase encoding becomes a bijection to its image. We describe the image of those restrictions using independent sets of graphs weighted with permutations. We derive the generating function for the independent sets and then for their weighted counterparts. The bijections we establish provide the enumeration of permutation classes. We use our results to uncover some unbalanced Wilf-equivalences of permutation classes and outline how to do random sampling in the permutation classes. In particular, we cover the classes $\mathrm{Av}(2314,3124)$, $\mathrm{Av}(2413,3142)$, $\mathrm{Av}(2413,3124)$, $\mathrm{Av}(2413,2134)$ and $\mathrm{Av}(2314,2143)$, as well as many subclasses.
Highlights
A permutation of size n is an arrangement of the numbers 1, 2, . . . , n
With each combination of patterns, we describe a set of patterns P that can be added to the basis while keeping the structural properties of the class that we need for enumeration
To this point we have considered permutation classes that can be described by filling the independent sets of the graphs U(Bn), D(Bn) and UDRC(Bn), which were first used by Bean et al (2020) to enumerate permutation classes avoiding size 3 patterns
Summary
A permutation of size n is an arrangement of the numbers 1, 2, . . . , n. The permutation σ avoids the pattern 54321 since it does not contain any decreasing substring of size 5. Bean et al (2020) introduced the staircase encoding, a function which maps a permutation to a staircase grid where cells are filled with non-negative integers. In this context, each integer is the size of the monotone sequence in its cell. Our notion of weighted independent sets is generalized to allow labelling
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